Fluctuations, large deviations and rigidity in hyperuniform systems: A brief survey

Ghosh, Subhroshekhar; Lebowitz, Joel L.

doi:10.1007/s13226-017-0248-1

Cited by 61 publications

(70 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Secondly, our findings are at variance with the behaviour observed in invariant models, as for instance the β-Gaussian, β-Wishart or β-Cauchy ensembles, in which there is no intermediate regime and, instead, the typical (Gaussian) and the large deviation regimes match smoothly [10,31]. We hope that our exact results for the FCS in the complex Ginibre ensemble will be interesting for the rather wide community working on the FCS in matrix models and related Coulomb gas systems, and more generally on random or disordered hyperuniform systems [21,22]. Besides, the results found in the present work could be experimentally relevant, as it has recently been shown that the positions of the eigenvalues of the complex Ginibre ensemble (2) are in one-to-one correspondence with the positions of fermions in a 2d-rotating harmonic trap [41], a system which, in principle, can be realized experimentally (see e.g.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble

et al. 2019

View full text Add to dashboard Cite

We study the Ginibre ensemble of N ×N complex random matrices and compute exactly, for any finite N , the full distribution as well as all the cumulants of the number Nr of eigenvalues within a disk of radius r centered at the origin. In the limit of large N , when the average density of eigenvalues becomes uniform over the unit disk, we show that for 0 < r < 1 the fluctuations of Nr around its mean value Nr ≈ N r 2 display three different regimes: (i) a typical Gaussian regime where the fluctuations are of order O(N 1/4 ), (ii) an intermediate regime where Nr − Nr = O( √ N ), and (iii) a large deviation regime where Nr − Nr = O(N ). This intermediate behaviour (ii) had been overlooked in previous studies and we show here that it ensures a smooth matching between the typical and the large deviation regimes. In addition, we demonstrate that this intermediate regime controls all the (centred) cumulants of Nr, which are all of order O( √ N ), and we compute them explicitly. Our analytical results are corroborated by precise "importance sampling" Monte Carlo simulations.arXiv:1904.01813v1 [cond-mat.stat-mech]

show abstract

Section: Discussionmentioning

confidence: 99%

“…The fact that the variance grows like ∝ r, i.e. much slower than the area of the disk (∝ r 2 ) demonstrates the hyperuniformity of the fluctuations in this system [21,22]. Besides, a more detailed analysis of the CGF allows to show that all the cumulants, other than the first one, grow at most like √ N [41].…”

Section: A Regime Of Typical Fluctuationsmentioning

confidence: 90%

Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble

et al. 2019

View full text Add to dashboard Cite

show abstract

“…Pair correlation functions g2(x) of the URL model in 1D, cf. Eqs (7). and (C6), where the random displacement of each point in the lattice Z is uniformly distributed in [−a/2, a/2).…”

mentioning

confidence: 99%

Cloaking the underlying long-range order of randomly perturbed lattices

2020

View full text Add to dashboard Cite

Random, uncorrelated displacements of particles on a lattice preserve the hyperuniformity of the original lattice, that is, normalized density fluctuations vanish in the limit of infinite wavelengths. In addition to a diffuse contribution, the scattering intensity from the the resulting point pattern typically inherits the Bragg peaks (long-range order) of the original lattice. Here we demonstrate how these Bragg peaks can be hidden in the effective diffraction pattern of independent and identically distributed perturbations. All Bragg peaks vanish if and only if the sum of all probability densities of the positions of the shifted lattice points is a constant at all positions. The underlying longrange order is then 'cloaked' in the sense that it cannot be reconstructed from the pair correlation function alone. On the one hand, density fluctuations increase monotonically with the strength of perturbations a, as measured by the hyperuniformity order metric Λ. On the other hand, the disappearance and reemergence of long-range order, depending on whether the system is cloaked or not as the perturbation strength increases, is manifestly captured by the τ order metric. Therefore, while the perturbation strength a may seem to be a natural choice for an order metric of perturbed lattices, the τ order metric is a superior choice. It is noteworthy that cloaked perturbed lattices allow one to easily simulate very large samples (with at least 10 6 particles) of disordered hyperuniform point patterns without Bragg peaks.

show abstract

“…All perfect crystals and quasicrystals are hyperuniform, but typical disordered many-particle systems, including gases, liquids, and glasses, are not. Disordered hyperuniform many-particle systems are exotic states of amorphous matter that have attracted considerable recent attention in physics and materials science because of their novel structural and physical properties [10][11][12][13][14][15][16][17][18][19][20]. According to the celebrated Riemann hypothesis, the nontrivial zeros of the zeta function lie along the critical line s = 1/2 + it with t ∈ R in the complex plane and thus form a one-dimensional point process.…”

Section: Introductionmentioning

confidence: 99%

Uncovering multiscale order in the prime numbers via scattering

2018

View full text Add to dashboard Cite

The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call effectively limit-periodic. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor S(k), proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor for primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. We identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes.

show abstract

Fluctuations, large deviations and rigidity in hyperuniform systems: A brief survey

Cited by 61 publications

References 39 publications

Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble

Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble

Cloaking the underlying long-range order of randomly perturbed lattices

Uncovering multiscale order in the prime numbers via scattering

Contact Info

Product

Resources

About